#512452

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

738 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,20,26)(15,22,27,39)(16,21,30,36)(17,24,33,42)(18,23,34,41)(19,35,32,25)(28,40,43,38)(29,44,31,37);(6,9)(10,12)(13,15,18,28,19,29,33,30,20,27,34,43,32,31,17,16)(14,21,24,37,25,38,41,39,26,36,42,44,35,40,23,22);(1,2,3,4)(15,27)(16,30)(21,36)(22,39)(28,43)(29,31)(37,44)(38,40);(5,7,8,11)(6,10,9,12)(13,16,17,31,32,43,34,27,20,30,33,29,19,28,18,15)(14,22,23,40,35,44,42,36,26,39,41,38,25,37,24,21)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=e,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=ghi,h^a=hi,c^b=ci,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419648);